Simply put, for Roamio to work as MoCA bridge, you have to connect the ethernet port on Roamio to your router
If you are only using Roamio as a MoCA client, then you don't need to hook up the ethernet port

I believe the answer is yes. If a MoCA network already exists, the Roamio (Plus or Pro model only) Ethernet port can be used to provide a connection for another device. I'm not 100%, but am fairly confident I've read posts from other users doing this. I would expect a switch would also work, but can't confirm .

Simply put, for Roamio to work as MoCA bridge, you have to connect the ethernet port on Roamio to your router
If you are only using Roamio as a MoCA client, then you don't need to hook up the ethernet port

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I don't understand why connecting the coax side to the router won't work.

From my understanding, if Roamio can get IP from router in MoCA mode, the ethernet port on Roamio can be used to connect other device either directly (one device) or to a switch then to multiple devices